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Electric Power Allocation in a Network of FastCharging Stations

I. Safak Bayram,Graduate Student Member, IEEE,George Michailidis,Member, IEEE,Michael Devetsikiotis,Fellow, IEEE,and Fabrizio GranelliSenior Member, IEEE

Abstract—In order to increase the penetration of electricvehicles, a network of fast charging stations that can providedrivers with a certain level of quality of service (QoS) is needed.However, given the strain that such a network can exert on thepower grid, and the mobility of loads represented by electricvehicles, operating it efficiently is a challenging and complexproblem. In this paper, we examine a network of charging stationsequipped with an energy storage device and propose a schemethat allocates power to them from the grid, as well as routescustomers. We examine three scenarios, gradually increasingtheir complexity. In the first one, all stations have identicalcharging capabilities and energy storage devices, draw constantpower from the grid and no routing decisions of customersare considered. It represents the current state of affairs andserves as a baseline for evaluating the performance of theproposed scheme. In the second scenario, power to the stations isallocated in an optimal manner from the grid and in addition acertain percentage of customers can be routed to nearby stations.In the final scenario, optimal allocation of both power fromthe grid and customers to stations is considered. The threescenarios are evaluated using real traffic traces correspondingto weekday rush hour from a large metropolitan area in the US.The results indicate that the proposed scheme offers substantialimprovements of performance compared to the current mode ofoperation; namely, more customers can be served with the sameamount of power, thus enabling the station operators to increasetheir profitability. Further, the scheme provides guarantees tocustomers in terms of the probability of being blocked (andhence not served) by the closest charging station to their location.Overall, the paper addresses key issues related to the efficientoperation, both from the perspective of the power grid and thedrivers satisfaction, of a network of charging stations.

Index Terms—Electric Vehicles, Stochastic Charging StationModel, Performance Evaluation

I. I NTRODUCTION

Over the last few years a strong push is occurring toreduce the use of hydrocarbons in transportation. This trendis supported by the latest advances in battery and convertertechnology, along with government mandates on energy inde-pendence and resilience and is enabled by the introduction of

Manuscript received 8 October 2012; revised 18 March 2013. Part of thework appeared in [1] at Smartgridcomm’12 Conference, Tainan City, Taiwan.

I. Safak Bayram and Michael Devetsikiotis are with the Departmentof Electrical and Computer Engineering, North Carolina State University,Raleigh, NC, 27695-7911 USA.

George Michailidis (corresponding author) is with the Departments ofStatistics and EECS, University of Michigan, Ann Arbor, MI,48109-1107,USA.

Fabrizio Granelli is with the Department of Information Engineering andComputer Science, University of Trento, Trento, Italy.

Emails:(isbayram, mdevets)@ncsu.edu, gmichail@umich.edu andgranelli@disi.unitn.it.

electric vehicles (EVs) and their close relatives Plug-in HybridElectric Vehicles (PHEVs) by major car manufacturers thathave drastically increased consumer choices [2], [3]. Althoughthere are diverging forecasts about the growth rate of the EVpopulation [4], there is consensus that it is going to representa sizable portion of the US fleet by 2025 - 30. Obviously,penetration rates could be significantly higher than these esti-mates depending on battery costs, gasoline prices, governmentpolicies, and the availability ofcharging infrastructure.

Indeed, such infrastructure is mostly needed in metropolitanareas, primarily characterized by higher population density,and where residents living in multi-unit dwellings do nothave easy access to night-time charging capabilities. A recentsurvey among EV drivers in California shows that 40% ofthem travel daily farther than the range of their fully chargedbattery [5], thus requiring a recharge during daytime operationof the vehicle. A network of fast charging stations overcomesthis problem [4].

On the other hand, there is concern about the strain that arapid adoption of EVs would exert on the power grid, due tothe large load that they represent [6]. Obviously, the extentof their impact will depend on the degree and local/regionaldensity of the EV penetration rate, charging requirement andthe time of the day they are charged. Nevertheless, deployinglarge scale charging stations may lead to grid instabilities.However, equipping each station with an energy storage devicecan reduce the impact of EV charging as shown in [7], [8].

The previous discussion indicates that efficient operationalregimes for a network of charging stations need to be de-veloped, so that they minimize the strain on the power grid,while at the same time offering good quality of service to EVdrivers. The aim of this study is to address these issues in acomprehensive manner. Specifically,

• We introduce an EV fast DC charging station architecture,introduce a stochastic model to capture its operationalcharacteristics and evaluate its performance (defined asthe percentage of served customers). The charging stationis equipped with a local energy storage device that aidssmoothing the stochastic customer demand.

• We propose a resource allocation framework that meetsQoStargets at each station and minimizes the amount ofpower employed. This framework is evaluated under threedifferent scenarios motivated by examining actual traffictraces from the Seattle area that exhibit a non-uniformspatial distribution of vehicles trips. The three scenariosin increased complexity are: (i) no power or customerallocation to station occurs, the stations are identical

2

in nature and act as inert service points; (ii) powerresources are allocated to each station; and (iii) optimalpower resource allocation is complemented by customerrerouting to neighboring stations.

• To achieve allocation of customers, a two-way communi-cations protocol is introduced that coordinates EV assign-ments and reroutes if necessary. The latter represents anessential element for increasing the number of EVs beingcharged with the same amount of power drawn from thegrid.

The remainder of this paper is organized as follows: Sec-tion II discusses related literature. In section III, we intro-duce our single charging station stochastic model, while insection IV, we introduce the network and present our powerresource allocation framework for the aforementioned cases.In section V, we analyze the Seattle traffic traces to estimatetraffic arrival rates for different stations in the area underconsideration. Further, we employ ideas from response surfacemethodology to estimate a model of the performance metricof interest -probability of an EV arriving to a charging stationand being blocked from receiving service- as a function ofthe station’s charging capacity, speed of battery chargingandvehicles arrival rate. The resulting model is used to solve thevarious allocation problems introduced in section IV and theirresults are compared.

II. RELATED WORK

There has been increased interest on devising schemes thatefficiently schedule EV chargings, on developing architecturesfor charging station, and for organizing and operating a net-work of charging stations. The following paragraphs providea brief overview of related literature.

Most works on scheduling EV chargings, assume station-ary vehicles located at customer premises or large parkinglots. The proposed charging strategies can be classified intothe following two categories. In the first one, there is acentral authority (dispatcher) that to a large extent controlsand mandates charging rates, start times, etc. [9], [10], [11],[12]. System level decisions involve selecting the desiredstate of charge, charging intervals, etc. are taken so as tofinish all charging requests by a prespecified deadline (e.g.7 am). The main advantage of a centrally controlled chargingschedule is that it leads to higher utilization of grid resources,together with real time monitoring of operational conditionsacross the entire power system. The second category examinesdecentralized decision making by EV owners. Specifically,they select individual charging patterns based on the prevailingprice of electricity or on self-imposed deadlines. It eliminatesthe need for a third party controller (dispatcher) and complexmonitoring techniques. Since decisions are taken individually,game theoretic models, such as mean field games, potentialgames, and network routing games are used in these studies[13], [14], [15], [16].

As will be seen in section V, we consider spatially dis-tributed, our study uses a centralized decision making mech-anism for a subset of EVs.

Currently there are only a handful of studies on chargingstation design. From a pure power engineering perspective [17]

proposes a fast charging station architecture with a DC busdistribution system. The station is equipped with an energystorage unit to minimize the strain on the grid, and the sizingproblem was determined by Monte Carlo simulations accord-ing to average load. A similar station architecture was usedin [18], [19], but two different energy storage devices wereconsidered; a flywheel and a supercapacitor. A mechanismthat simultaneously draws power from the grid and the storagedevices was introduced to decrease the EVs charging duration.However, there is a multitude of storage technologies in themarket and the choice of the most appropriate one is mostlystation dependent (e.g. a low energy density, large size butinexpensive storage device may not be suitable for a stationlocated at or near city centers, due to real-estate costs) [1],[20]. Thus, in our station architecture we examine differentstorage technologies, characterized by their efficienciesandpower ratings.

III. C HARGING STATION ARCHITECTURE

The design of a network of charging stations is ultimatelylinked to the current power grid operations. At present, cus-tomer demand -household, commercial and industrial- can beassigned to three categories based on service costs. The firstrepresents the base load that is supplied by large, low cost (perkWh) generation assets, such as nuclear, coal and hydro. Largesize industrial customers with fairly steady demand, togetherwith an aggregate estimate of households and commercialusers belong to this category. The second category representsthe difference between base load generation and expected ag-gregate demand and is primarily met by gas/liquid fuel powerstations. Finally, the third category represents peak demandthat is met by fast start generators, which are characterized bytheir high cost (per kWh).

EVs represent sizable,mobileelectric loads. Level-1 charg-ing represents a load comparable to a household, while Level-2 charging a load twice as large to a household. Thus, largenumber of EVs, geographically concentrated, would imposehuge strains not only on power generation, but also on the griddistribution system [6], [21], [22]. In some studies [23], [24]it is argued that if just5% of all EVs charge simultaneously atfast charging stations,5 GW of extra power would be neededby year2018 in the VACAR region (Virginia - North Carolina- South Carolina). For these reasons, charging station designsthat do not stress the power grid and eliminate the need foradding significant extra generation capacity become important.

The four key components of our design that try to addressthese issues are: (I) each station draws constant power fromthegrid; (II) local energy storage is employed to meet stochasticcustomer demand; (III) the station supports different classes ofcharging requests (fast service vs slow service); and (IV) theQoS metric employed is the long-term blocking probability ofincoming customers. The overview of the proposed chargingstation is depicted in Figure 1. Next, we explain the systemdynamics in detail.

(I) Charging stations of any significant size represent com-mercial size loads. Hence, it seems reasonable for stationoperators to draw long-term contracts with the utility where

3

Utility Supplies

Constant Power

Stochastic

Demand is taken

care of from the

energy storage

Power Demand

...

POWER

GRID

s

r

...

Class-1

Class-c

Class-c

Class-1

Fig. 1: Single Charging Station Architecture

a power level is agreed in return for a lower price. Thisenables the utility to better anticipate its demand, and thestation operator to benefit from a lower price; as argued in[25], [26],such contracts leads to lower contract, as well asaverage spot prices, and more efficient market equilibria.

(II) Energy storage represents a critical component in theproposed system architecture, since it aids in smoothing cus-tomers’ stochastic demand. During rush hour, stored energycan be used to serve more customers. Similarly, when thepower that the station can draw from the grid is not fullyutilized, the extra power can recharge the storage device. Anoverview of candidate technologies for storage devices andtheir efficiencies are presented in Figure 2. Their details willbe further explained in the next section. (III-A).

(III) We consider that the charging station provides serviceto multiple customer classes at different charging rates. Thisallows the station to accommodate customers with differentcharging needs and preferences, as well as EVs with differenttechnological constraints.

(IV) As discussed in the introduction, charging times dependon the level used, but on average they are about 30 minutes.Hence, it is reasonable to assume that incoming customerswould not be willing to wait and thus in our model (discussednext) a “bufferless” system was adopted. For such a system,the blocking probability becomes a natural performance met-ric.

A. Stochastic Model for Station Dynamics

Based on the aforementioned specifications, the proposedstation architecture and the corresponding model for its behav-ior over time, exhibit the following operation characteristics:(i) the charging station draws a constant power from the grid;(ii) upon exceeding the available grid power, the local energystorage unit is used to meet additional demand; (iii) when-ever there is idle grid power, it is used to charge the localenergy storage device, if it is not in a fully charged state;(iv) depending on the amount of constantly drawn grid powerand the size of the local energy storage, a certain level ofQoS is provided; and (v) the station partitions its capacitywith respect to demand for each customer class. Such insightscan be obtained from profiling studies (e.g. customer surveys,etc). The constantly drawn grid power is discretized toS equal

Technology Efficiency

Battery 60%-80%

Flywheel 93%

Sup-capacitors 95%

Hydrogen Fuel Cell 59%

Fig. 2: Candidate Energy Storage Systems Technology Land-scape [27] [28]

slots, meaning that it can accommodate up toS vehicles atthe same time. In a similar way, the local energy storage canchargeR vehicles in a fully charged state. Since the chargingstation can never serve more thanS+R vehicles concurrently,the very next EV arrival is going to be “blocked”. This strategyinsulates to a large extent the power grid from peak demand.The details of the stochastic model are given next. Customersarrive to the charging facility according to a Poisson processwith parameterλ. Currently, a variety of different EV modelswith different battery sizes exist. Thus, the service time ofcustomers is assumed to be a exponentially distributed withrateµ. Also, the charging duration of the energy storage deviceso that it is able to accommodate one more EV is exponentiallydistributed with rateν which depends on the underlying energystorage technology.

We proceed to explain the energy storage component in de-tail. The power rating of an energy storage device determineshow fast it can be charged. Thus, energy stored in unit timecan be calculated by the product of the power rating and theefficiency (ratio of stored energy and total amount of energyspent to charge battery) of the energy storage unit. This meansthat depending on these two parameters, different amountsof energy can be stored. For instance, assume that our fastcharging station can charge an EV (battery withη = 0.9) in30 minutes using maximum power rating,SPR = 1 (µ = 2).Also, we employ an energy storage with the same efficiency,but with a higher power ratingSPR = 2 than the EV battery,so that in the same amount of time, we can store up the energyfor the demand coming from 2 EVs in the local storage device(ν = 4). Note that the charging rate isν = f(S, η), withS ≤ SPR being the available power.

Given the assumptions above, the single charging sta-tion model can be represented by continuous time Markovchain with 2-dimensional finite state space. In Figure 3,the state space of the Markov chain, along with its trans-mission rates are depicted. The total number of states is

κ = (S+1)(R+1)+R∑

i=1

i. It is easy to see that a unique

steady state distribution would exist which can be calculatedby solving:

πQ = 0 and πe= 1 (1)

wheree is a column vector whose elements are all equal to 1,andQ is a κ× κ matrix containing the transition rates andπa vector of lengthκ containing the steady state probabilities.

4

BL

OC

KIN

GST

AT

ES

0,0 1,0 2,0 S,0S-

1,0

S+2,

R-1

S+2,

2

S+1,

1

S+R,

R

µ 2µ 3µ (S-1)µ Sµ

0,1 1,1 2,1 S,1S-1,1

µ 2µ 3µ (S-1)µ Sµ

…..

…..

S+1,

20,2 1,2 2,2 S,2S-1,2

µ 2µ 3µ (S-1)µ Sµ

…..

Sµ

Sµ Sµ

v

v v v v

v v v

µ

2µ

v………..

………..

………..

………..

………..

………..

3µv v v v

µ

S+1,

R-1

0,

R-1

1,

R-1

2,

R-1

S,

R-1

S-1,

R-1

µ 2µ 3µ (S-1)µ Sµ

…..

Sµ

S+R-1,R-1…..

Sµ

v v v v

µ 2µ

µ2µ

(R-1)µ

….

….

….

….

S+2,

RS+1,R

0,R 1,R 2,R S,RS-1,R

µ (S-1)µ Sµ

…..

Sµ

S+R-1, R…..

Sµ Sµ Sµ

v v v v

Sµ

Rµ(R

-1)µ

3µ2µµ

1,0 2,0 S,0S-1,0

0,2 1,2 2,2

0,0 i: Number of Cars in the

Charging Station

j: Local Energy Storage

Charge Level

2µ 3µ

Sµ Sµ

Energy Storage States

Grid Charging States

Fig. 3: Continuous Time Markov Chain

Note that the elements ofQ satisfy qab ≥ 0 for a 6= b andqab = −

∑κa=1,b6=a qab for all a = 1, 2, . . . , κ. Then, the

model’s (station’s) blocking probability can be calculated fromS∑

i=1

π( i(i+2S+1)2 ), i = 1 . . . S.

Q =

−(λ+ ν) λ · · · 0µ −(λ+ ν + µ) · · · 0...

.... . .

...0 0 · · · −(S +R)µ

(2)

Next, we extend this model to the case where differentclasses of customers are present; namely,c ∈ {1, · · · , C}.Also, denote by~ρ the percentage of customers that demandclass-c type of service. Then, the station operator partitionsthe power drawn from the grid intoC components by solvingthe following optimization problem.

argminS(c)

∑

c∈C

B(c)(~ρλ, S(c), R(c))

s.t.∑

c∈C

S(c) = S

~R(c), ~ρ, and λ are given

(3)

To illustrate how the characteristics of the energy storagedevice improve performance of the station we use the follow-ing example. We fix the size of two devices, but we vary theirefficiency and power rating parameters. There is a fast energystorage with95% efficiency andSPR=2, and a slow one withefficiency of85% andSPR=1. Storage size is set toR=5 andthe EV arrival rate varies between (λ = 1 − 7). To ease thedemonstration, a single customer class is assumed requestinga charging rate ofµ=2. As shown in Figure 4, the fast energy

storage device outperforms the slow one in terms of blockingprobabilities.

Next, we evaluate the system performance (percentage ofvehicles it can charge), under the following sets of parameters.There are two customer classes; in class-1 EVs request fastcharging, while in class-2 request slower charging. A typicalcharging duration takes30 minutes, thus the charging rateµ(1)

is set to2 and µ(2)=1. We assume that the station operatorpicks the energy storage according to the following specifica-tions (note that superscript denotes the customer class): storagesize R(1)=R(2)=5, efficiency η(1)=0.95 and η(2)=0.85 andpower ratingsS(1)

PR = 2 andS(2)PR=1. Based on an EV profiling

study, it is estimated that the total arrival rates varies betweenλ= 1−7. We look at three different compositions of the EV pop-ulation: (ρ1, ρ2) = {(75%, 25%), (50%, 50%), (25%, 75%)}.Then, the station operator solves optimization problem 3 tocalculate the optimal~S(c) given by[6, 4], [4, 6], and[2, 8] forthe given(ρ1, ρ2) pairs, respectively. The resulting blockingprobabilities are shown in Figure 5. It can be seen that thesystem can serve more customers, in the presence of a largerpercentage of fast charging customers. This is expected, sincethe overall “service rate” is faster in that case.

B. Profit Model

The previous performance assessment provides insight intothe gains captured by the posited QoS, namely the blockingprobability. Next, we present a charging station profit modelthat relates the stochastic model to cost parameters. This modelprovides guidance to choose the right values for the amountof power drawn from the grid for different arrival rates.

The principles of the profit model forC different customerclasses are as follows: the charging station earns differential

5

1 2 3 4 5 6 70

0.005

0.01

0.015

0.02

0.025

Arrival Rate λ

Blo

ckin

g P

rob

abil

ity

Fast Energy Storage

Slow Energy Storage

Fig. 4: Performance Evaluation of Different Energy StorageDevices

1 2 3 4 5 6 70

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Arrival Rate λ

To

tal

Blo

ckin

g P

rob

abil

ity

T

wo C

lass

es

Class 1 (fast) = 75%, Class 2 (slow) = 25%

Class 1 (fast) = 50%, Class 2 (slow) = 50%

Class 1 (fast) = 25%, Class 2 (slow) = 75%

Fig. 5: Multi Class Charging Station Performance Evaluation

revenue for each served EV according to its class (e.g. morerevenue from fast charging customers etc.). On the otherhand, a penalty is paid for each blocked EV because (1) itleads to dissatisfied customers and degrades the reputationof the station; (2) it enables to control the QoS to fosterEV adoption [4]; (3) it allows station operators to size itscapacity to maximize its profit. It is assumed that a higherpenalty is paid to customers charged more for service. LetR

(c)g and R

(c)l be the revenue gained per EV class-c, when

served from the grid and the energy storage, respectively.Further, letC(c)

b denote the blocking cost of a single EVin class-c. Finally, let C0 represent the fixed installation costand C

(c)a R the acquisition cost, assumed to be proportional

to size, for customer class-c of the storage unit. In order tocalculate the net profit, for each customer class, we classifythe charging states in the Markov chain model to: the “gridcharging states” and the “storage unit charging states”. Letρ(g) = {(i, j) : 0 ≤ i ≤ S, 0 ≤ j ≤ R} denote ‘the grid charg-ing states” andρ(l) = {(i, j) : S+ j ≤ i ≤ S+R, 1 ≤ j ≤ R}“the storage unit charging states”. Similarly,ρ(bl) representsthe “blocking states”, whilei(s) denotes the number of EVsat states. Then, the proposed profit function can be writtenas

P =∑

c∈C

∑

s∈ρ(g)

R(c)g i(c)(s)π(c)(s) +

∑

c∈C

∑

s∈ρ(l)

R(c)l i(c)(s)π(c)(s)

−(C0 +∑

c∈C

R(c)C(c)a )−

∑

c∈C

∑

s∈ρ(bl)

C(c)b i(c)(s)π(c)(s) (4)

We evaluate the profit model for the following set ofparameters in the presence of two customer classes (fast/slow):R

(1)g =R(1)

l =3, R(2)g =R(2)

l =1.5, C(1)b =3.5, C(2)

b =2, C(1)a =0.25,

0 5 10 155

0

5

10

15

20

Arrival Rate λ

Net

Pro

fit

Class 1 (fast)=75%, Class 2 (slow) =25%

Class 1 (fast)=50%, Class 2 (slow) =50%

Class 1 (fast)=25%, Class 2 (slow) =75%

Fig. 6: Multi Class Charging Station Net Profit

C(2)a =0.15 andC0=0.02. The results are shown in Figure 6. For

low arrival rates, the cost related to acquisition and installationoutweighs the revenue gained from charging EVs, and hencea negative profit is earned. On the other hand, for high arrivalsrates, the cost of blocking customers becomes dominant andthe total net profit decreases. Moreover, since fast charginglowers the blocking probability, the system means more profitwhen the proportion of class-1 customers is higher.

IV. A N ETWORK OFCHARGING STATIONS

A. Overview

Fast public charging stations are key to build confidence inthe early stages of EV adoption. At present, the number of fastcharging stations in the US is quite low, and deployment plansin the short term are limited to selected highways only [29],[30]. In order to compete against gas stations, deploying urbancharging facilities becomes necessary [31]. In this section,the operation of a network of fast charging stations in anurban environment is studied, where each individual station ismodeled according to the architecture introduced in section III.

In the real world, urban traffic movements are far frombeing uniform. In fact, people drive between specific pointsofinterest, such as their home, school, workplace, etc. Drivingpatterns vary according to the time of the day (weekday rushhours, weekends etc.) and hence traffic density represents adominant factor in the utilization of each node in a chargingstation network. As the power grid limitations prevent stationsfrom providing more capacity, grid operators have to considerthe fact of spatial and temporal demand to optimally allocatetheir power resources.

B. Power Resource Allocation in a Charging Station Network

1) Case-I: No Allocation: In the first case, all chargingstations in the network are assumed to be identical. Letl = 1, 2, ..., N be the index set of charging stations. Furtherassume that each station servesc ∈ C types of customerclasses, so thatS(c)

1 =S(c)2 . . . S

(c)N andR(c)

1 = R(c)2 · · · = R

(c)N .

The only parameter that differs in these stations is the arrivalrateλi and composition of the customer class populations~ρ,which comes from the traffic density (note that we considerrational customers who always drive to the nearest station).

6

2) Case-II: Optimal grid power-S allocation within a largegeographical urban areas:Similarly to the case above, thereareN charging stations deployed in a large urban environment.However, customers also have access to charging stationlocation information provided by a central authority1. Thiscase is divided into two subcases. The first subcase assumesthat all drivers are selfish and similarly to Case I, they choosethe nearest charging station. The second subcase assumesa hybrid population of selfish drivers and EV fleets. Notethat unlike selfish users, EV fleets adhere to the decisionsof the power utility to fulfill the requirements of customeragreements. Hence, the arrival rate of each station can beshaped within a[λmin, λmax] range.

Let Smax be the maximum level of generation capacity thatthe grid can supply to the network in a metropolitan area. Alsoeach station servesc ∈ C types of customer classes. Using thediscretization assumption at each charging node, two resourceallocation problems are formulated as a mixed integer non-linear programming problem in Equations 5 and 6. For bothsubcases, the proposed scheme allocates more power resourcesto the busier stations, while taking into accountQoS targets.If the total power required to satisfy theQoSrequirements isgreater thanSmax, then the charging station network providesbest-effort service with the maximum allowable grid power,Smax.

a) :

minS

∑

i∈l

∑

c∈C

Bi(~ρiλi, S(c)i , R

(c)i )

s.t.∑

i∈l

∑

c∈C

S(c)i = S

0 ≤ Bi(~ρλi, S(c)i , R

(c)i ) ≤ ǫ

S(c)i ∈ Z

+

R(c)i , λi, and ~ρi, are given

∀i ∈ l, ∀c ∈ C

(5)

b) :

minS,λ

∑

i∈l

∑

c∈C

Bi(~ρiλi, S(c)i , R

(c)i )

s.t.∑

i∈l

∑

c∈C

S(c)i = S

0 ≤ Bi(~ρλi, S(c)i , R

(c)i ) ≤ ǫ

λ(c)min ≤ λ

(c)i ≤ λ(c)

max

S(c)i ∈ Z

+

R(c)i , ~ρi , λ

(c)min and λ(c)

max are given

∀i ∈ l, ∀c ∈ C

(6)

3) Case-III: Optimal S and λ allocation in small geo-graphical areas: Let l∗ ⊂ l and 0 < n ≤ N . In thiscase, a charging station network deployed over a relativelywell confined small geographical area withn stations isconsidered. This case is different from the previous scenario inthe following aspect: the total population consists of EV fleetsand through agreements, customers can be assigned to any

1Via smart apps such as [32] or on board communication systems[33]

neighboring station. Since the considered distances betweenstations are reasonably short (2−3 miles2), routing customersto other stations would have negligible cost to drivers. Thus,customers can be assigned to neighboring area stations tominimize the total blocking probability. In all cases, the localenergy storage is assumed to have already been acquired bythe charging station(e.g.R = 5), thus its size is fixed. Finally,each station servesc ∈ C classes of customers, and routedcustomers get the same type of service. Then, the optimizationproblem becomes:

minS,λ

∑

i∈l

∑

c∈C

Bi(~ρiλi, S(c)i , R

(c)i )

s.t.∑

i∈l

∑

c∈C

S(c)i = S

∑

i∈l∗

λi = λ

0 ≤ Bi(~ρiλi, S(c)i , R

(c)i ) ≤ ǫ

λ(c)i ≥ 0, ∀i ∈ l∗

S(c)i ∈ Z

+

Ri and ~ρi are given

∀i ∈ l∗, ∀c ∈ C

(7)

In addition to the system constraints presented for each alloca-tion problem, there may be additional constraints, dependingon the existing power network, such as distribution networklimitations, etc. However, since the interaction of the chargingstations with the grid is limited to the constant power drawnfrom it, these case-by-case varying constraints may only affectthe maximum power allocation for individual stations. Thus,these constraints can easily be incorporated within the existingformulation to address these allocation problems.

V. EVALUATION & RESULTS

A. Overview

Collecting vehicular traffic traces, especially in urban areas,is a challenging and costly task. Hence, vehicles movementsare not well calibrated. However, in [34] bus movements fromthe Seattle area were obtained. Due to the city’s physicallayout and extensive bus network3, it is claimed that thesemovements resemble actual traffic patterns quite closely. In thenext subsection (V-A1), we use this publicly available datatoinvestigate the spatial distribution of vehicles, during weekdayrush hour (7am-9am and 5pm-7pm). The remainder of thissubsection is organized as follows. In subsection V-A2, weexplain our methodology in locating fast charging stationsonthe city map. In subsection V-B1, we use the Response SurfaceMethodology to approximate charging station blocking prob-abilities into a second order regression metamodel. Finally, insubsection V-C, we solve the optimization problems presentedin section IV-B using our metamodel.

2It would require0.5 − 1kWh of stored energy and would cost10-20cents with the current rates.

31200 buses in a5000 square kilometers area

7

Fig. 7: Fast DC charging station map in Seattle, WA [34]

1) Input Analysis:According to [34], the location of eachbus was recorded frequently. We start by normalizing thexandy coordinates of the input data. Subsequently, the ARENAInput Analyzer [35] is used to fit a spatial distribution to thedata. The results indicated that with mean squared error of0.6%, the spatial distribution of vehicles is a piecewise betadistribution for weekday rush hours. The results are presentedin equations 8 and 9.

f(X) =

{

44×BETA(4.42, 0.763) 0 ≤ X ≤ 44

44 + 137×BETA(0.752, 4.7) 44 ≤ X ≤ 180(8)

f(Y ) =

{

150×BETA(2.42, 0.799) 0 ≤ Y ≤ 150

150 + 121×BETA(1.07, 5.44) 150≤ Y ≤ 270(9)

In addition, we analyze the correlation ofx andy coordi-nates, and calculate the correlation coefficient as0.06.

2) Charging Station Placement:In [34], researchers placedeight base station towers in such a way that base stations cancommunicate with all mobile nodes. Since the charging stationlayout problem is outside the scope of this paper, a similarapproach is used and the same number of charging stationsis deployed in the same locations given by the followingcoordinates:{xi, yi} = {60, 45}, {60, 90}, {60, 135},{60, 180}, {60, 225}, {100, 90}, {100, 160}, {100, 225}.Figure 7 presents the map with the locations of the chargingstations. In order to calculate the traffic intensity at eachstation, a discrete event simulation model is used. We presentits flowchart in Figure 8. The station parameters are given byS = 5, R = 5, µ = 2, ν = 4 (assuming only fast chargingcustomers) for all stations. The simulation is terminated whenone million vehicles get serviced. It is run for a total of30 times and95% confidence intervals of the parameters ofinterest are obtained. The traffic intensity for each station is

Initialize

Simulation

Next

Event

-Determine Charge Type

-Advance Clock-Update Power

Resources

n=0?

Schedule

Next

Departure

EV/PHEV Departure

NO

-Determine Charge Type

-Advance Clock-Update Power

Resources

Schedule

Next Arrival

n=1?NO

EV/PHEV Arrival

YES

Enough Power

Resources?

Local ESS Charge

NO

-Advance Clock

-Update Power

Resources

-Schedule Local

ESS ChargeEnd

Schedule

Next Local

ESS Charge

YES

Fig. 8: Discrete Event Simulation Flow Chart

TABLE I: Traffic Intensity (T.I.) of Each Station

Sta. ID mean(T.I.) 95% CI Sta. ID mean(T.I.) 95% CI

1 3.56% 0.030% 2 9.68% 0.022%3 36.99% 0.060% 4 36.68% 0.035%5 1.85% 0.037% 6 1.65% 0.018%7 8.6% 0.055% 8 0.98% 0.016%

shown in Table I. It can be seen from Table I that chargingstations three and four are used to meet most of the chargingdemand, whereas other stations have relatively little demand.For instance, letting the overall arrival rate beλ = 50, then theblocking probabilities for the eight identical stations would be~Bi = [0.019, 0.053, 0.58, 0.58, 0.0158, 0.0153, 0.043, 0.014]. Itcan be concluded from this expository calculation that, withoutany power allocation, there could be severe fluctuations interms ofQoSamong the charging facilities4.

B. Output Analysis

1) Metamodeling of Blocking Probabilities:In section III,numerical methods are used to calculate EV blocking proba-bilities. However, new calculations are needed for each setofnew input parametersS, R, ν, andλ to determine the block-ing probabilityB. using the Response Surface Methodology(RSM) we are able to calculate an approximate second orderpolynomial model for the functional relationship betweenBand the input parameters (B = f(S,R, ν, λ) [36]. As inputparameters we used those presented in Table II, keepingµ isfixed to 2. The handicap of this approach is that the blockingprobabilities have to be in the[0, 1] interval, whereas the RSMmodel can predict values outside it. For that reason, we fit the

4Some stations(e.g. station-3) will exhibit a very high blocking probability,whereas overprovisioned stations (e.g. station-8) will exhibit a very lowblocking probability.

8

TABLE II: RSM Input Parameters

Parameter Interval Increments Type

S [1,15] 1 IntegerR [1,15] 1 Integerλ [0.25,30] 0.25 Floatν [2,10] 1 Integer

RSM model to the logit transformation (y = log(x/(1− x)))of B and then use the inverse-logit (x = 1/(1 + e−y))transformation to obtain the final results. The regression modelwhere the response variable corresponds to logit(B) is givenin Equation 10.

B(S,R, λ, ν) = −3.990− 2.666S − 1.6152R− 0.1492ν

+3.840λ− 0.0645SR− 0.002Sν + 0.209Sλ− 0.0078Rν

+0.094Rλ+ 0.003νλ− 0.0175S2 + 0.055R2 + 0.0089ν2

−0.271λ2

(10)

Then, the blocking probability becomes,

Blocking Prob. =

{

B(·)0

ifλ > 0ifλ = 0

(11)

For the above regression model, the R-Square statistic is88.06% and the mean square root error is0.52%.

∂B∂S∂B∂R∂B∂ν

∂B∂λ

=

−0.035S − 0.0645R − 0.002ν + 0.21λ − 2.66

0.0014S + 0.11R − 0.008ν + 0.094λ − 1.62

−0.002S − 0.078R + 0.178ν + 0.025λ − 0.15

0.209S + 0.094R + 0.003ν − 0.54λ + 3.84

(12)

H =

−0.035 −0.0645 −0.002 0.210.0014 0.11 −0.008 0.094−0.002 −0.078 0.178 0.0250.209 0.094 0.003 −0.54

(13)

Some key quantities like the Jacobian (equation 12) andthe Hessian matrix (equaiton 13) are given to aid assessingthe sensitivity ofB with respect to inputsS, R, ν and λvariables is presented. It can be seen that grid powerS has thehighest impact for decreasing the blocking probability. Notethat during periods of high arrival rates, there is going to belittle spare capacity left and hence the local storage devicewould be frequently in an empty state, as indicated by theseresults.

C. Comparison of three cases

Next, we compare the performance of the following threescenarios: (i) all eight stations are identical (case-I); (ii) powerresource allocation for selfish EV population (case-IIA) andmixed (selfish and EV fleets) population (case-IIB); and(iii) power resource allocation for EV fleets only (case-III).

20

22

24

26

28

0

0.02

0.04

0.06

0.0820

30

40

50

20

30

40

∑

i∈l

λi

Minimum S required to reach ε−Blocking Prob.

ǫ

min

∑ i∈

l

Si

Fig. 9: Minimum grid power required to meetǫ QoStargets

Standard interior point methods are used to solve the optimiza-tion problems introduced in section IV-B. Problems formulatedin case-II and case-III are non-linear integer programs andtheyare solved by relaxing the integer constraint and ceiling tothenearest integer. For case-IIA, suppose that the station operatorwants to provideǫ-level QoSat all stations. One of the maingoals of this scheme is to use the minimum amount of powergrid resources (for illustration assume all customers demandfast charging). Hence, the minimum required grid powerSmin

to meet theQoStargets is calculated. As long asSmin ≤ Smax

where Smax is the total allocated generation capacity, thistarget is going to be reached. A generic calculation is presentedin Figure 9.

Next, let us compare cases-I and -IIA. Suppose that thecharging station operator wants to ensure that each stationcan meet90% of the customer demand at all times (ǫ=0.10).For the eight stations, the arrival rate is assumed to beλ =27. Since the majority of the population resides near Stations2, 3 and 4, we assume that these two stations serve twotypes of customers; class-1 (fast chargingµ=2) and class-2(slow charging µ=1). The same set of parameters fromsection III-A are used for the efficiency and the power ratingof the local energy storage units. Since these regions areclose to downtown we further assume that~ρ=(75%, 25%).The remainder of the stations serve customer class-1. Solvingequation 5 results in~S=[1, 2, 9, 9, 1, 1, 2, 1]. With theallocated grid power, blocking probabilities for each station are~B=[0.0094, 0.028, 0.099, 0.087, 0.0004, 0.0023, 0.016, 0.0001].In order to compare the performance of the whole network, wecalculate the weighted sum of stations’ blocking probabilities;

∑

i∈l

wiBi, wherewi =λi

∑

i∈l

λi

(14)

Then, the weighted sum of blocking becomes∑

i∈l

wiBi=0.0440. To compare these results with case I,

assume that each station hasSi = 3 (exceptS3 = S4 = 4)andR = 5. Arrival rates are the same as case II. For this case,the weighted sum of blocking becomes

∑

i∈l

wiBi=0.4365. This

9

TABLE III: Results for Case IIB (Mixed Population of Selfish EVs and Fleets)∑

i∈l

λi Station1 Station2 Station3 Station4

S1 λ1 B1 S2 λ2 B2 S3 λ3 B3 S4 λ4 B4

ǫ=0.

05 20 1 0.77 0.0037 2 2.656 0.015 5 7.3 0.0347 5 6.984 0.03225 1 0.9625 0.0153 3 2.832 0.0125 6 8.98 0.047 6 8.90 0.04130 1 1.155 0.032 3 3.3984 0.031 7 10.767 0.05 7 10.4760 0.046

ǫ=0.

10 20 1 0.77 0.057 1 2.2656 0.0317 4 7.178 0.10 4 6.98 0.093225 1 0.9625 0.0153 1 2.832 0.03491 5 8.9725 0.10 5 8.73 0.093630 1 1.155 0.032 2 3.3984 0.031 6 10.76 0.10 6 10.476 0.0934

∑

i∈l

λi Station5 Station6 Station7 Station8

S5 λ5 B5 S6 λ6 B6 S7 λ7 B7 S8 λ8 B8

ǫ=0.

05 20 1 0.3960 0.00122 1 0.3663 0.0007 2 1.892 0.0087 1 0.022 025 1 0.495 0.0007 1 0.4538 0.0004 2 2.365 0.0379 1 0.0275 030 1 0.5940 0.0017 1 0.7445 0.0011 3 2.84 0.0127 1 0.033 0

ǫ=0.

10 20 1 0.396 0.0002 1 0.3630 0.0001 2 1.892 0.014 1 0.022 025 1 0.795 0.0007 1 0.4538 0.0004 2 2.365 0.0379 1 0.1875 030 1 0.594 0.0017 1 05445 0.0011 2 2.883 0.0764 1 0.651 0.001

TABLE IV: Results for Case-IIA (Selfish EVs)∑

i∈l

λi Station1 Station2 Station3 Station4

ǫ=0.

05

S1 B1 S2 B2 S3 B3 S4 B4

20 1 0.0023 2 0.0094 6 0.025 6 0.021425 1 0.0067 2 0.0269 7 0.0348 7 0.029930 1 0.0148 3 0.0082 8 0.0436 8 0.0376

ǫ=0.

10 20 1 0.023 2 0.0094 5 0.0641 5 0.056825 1 0.0067 2 0.0269 6 0.0754 6 0.066930 1 0.0148 2 0.0568 7 0.0842 7 0.0748∑

i∈l

λi Station5 Station6 Station7 Station8

ǫ=0.

05

S5 B5 S6 B6 S7 B7 S8 B8

20 1 0.0001 1 0 2 0.0054 1 025 1 0.0002 1 0.0002 2 0.063 1 030 1 0.0006 1 0.0004 2 0.0367 1 0

ǫ=0.

10 20 1 0.0001 1 0 2 0.0911 1 025 1 0.0002 1 0.0002 2 0.0163 1 030 1 0.0006 1 0.0004 2 0.0367 1 0

TABLE V: Comparison of Case-IIA (Selfish EVs) and Case-IIB (Mixed Population)

∑

i∈l

λi SCase IIAi

SCase IIBi

Savings

ǫ=0.

05 20 20 18 10%25 22 21 4.55%30 25 24 4%

ǫ=0.

10 20 18 15 18.75%25 20 17 15%30 22 20 9%

TABLE VI: Results for Case III (EV Fleets)∑

i∈l∗

Si

∑

i∈l∗

λi Station2 Station3 Station4

S2 λ2 B2 S3 λ3 B3 S4 λ4 B4

18 16.67 6 5.56 0.004 6 5.56 0.004 6 5.56 0.00424 20.83 8 6.94 0.0327 8 6.94 0.0327 8 6.94 0.032730 24.9 10 8.3 0.005 10 8.3 0.005 10 8.3 0.005

02

46

810

0

5

100

0.5

1

Grid Power-S

X: 5Y: 5Z: 0.07527

Arrival Rate-λ

∑ i∈

l

Bi

Fig. 10: Evaluation of Equation 7

sample calculation shows that with power resource allocation,more vehicles can receive service with the same amount ofgrid power.

For the allocation problem in cases II-B and III, a two-way communication infrastructure is used to offer customersincentives to charge from other stations. In the first case, acentral authority can route a certain percentage of customersin the [λmin, λmax] range. In the latter one, any customer canbe assigned to any station in the same neighboring area. Thus,for the first case suppose that the arrival rate of each station isin theλ±10% range. For instance, if station-3 the arrival rateis λ3 = 10 arrival rate will be assigned in [9,11] interval. Next,two allocation problems 5 and 6 are solved for six differentcombinations:ǫ = 0.05, 0.10 and

∑

i∈l

λi = 20, 25, and30.

We proceed to compare three cases for all stations in detail.Table IV presents the results for a population of selfish EVusers. The utility can only allocate optimal power (problem5).On the other hand, for a mixed population of EVs (selfish andFleets) allocation problem 6 is solved. We present detailedresults for each charging facility in Table III. Note that sincethe central authority can partially affect the customer choices,blocking probability targets can be achieved with less gridpower. For instance, customer routing can lead to10% powersavings to provideǫ=0.05 QoS. Detailed results are given inTable V. Moreover, the comparison of these two cases andassociated savings are presented in Table V.

10

Arrv. Rate=20 Arrv. Rate=25 Arrv. Rate=300

0.1

0.2

0.3

0.4

0.5W

eig

hte

d S

um

of

To

tal

Blo

ckin

g P

rob

abil

ity

Case I

Case IIA

Case IIB

Case III

Fig. 11: Comparison of three cases respect to stations 2, 3 and4

Case-III assumes a population of pure EV fleets. Notethat the network map (Figure 7) is divided into smallergeographic areas, and inside each region the cost of drivingbetween charging stations has a negligible cost. We assumethat charging stations2, 3, and4 constitute a charging network.Similar to the previous case, a central authority through theuse of two-way communications, can assign customers to anystation in this subarea. To evaluate this case, assume thatstations2, 3, and4 are in a small well-confined neighborhoodand driving between these stations has a negligible cost. Then,in optimization problem 7, minimum customer blocking prob-abilities are obtained atSi=S/N andλ

(1)i =λ(1)/N whereN

is the number of charging stations (alsoR(1)1 = R

(1)2 . . . R

(1)N

andR(2)1 = R

(2)2 . . . R

(2)N )5. We present the results forǫ = 0.05

and varying arrival rate parameters in Table VI. Moreover, werun a sample calculation for two stations, with the followingparameters:

∑

i∈l

λi = 10,∑

i∈l

Si = 10 andR1 = R2 = 5. The

results are shown in Figure 10.Next, we compare the baseline scenario (no allocation of

any kind) and three allocation schemes for stations2, 3, and4since they serve both fast and slow customer classes. For a faircomparison, we fixed the total grid resources, and employ thesame type of energy storage devices. Results are depicted inFigure 11 for three different arrival rates. In order to quantifythe effects of power allocation and customer routing on thecharging network, the profit model of section III-B is appliedto all stations. Previously presented results for all threecasesare used forǫ = 0.05 and arrival ratesλ = 20, 25, and 30.The same set of parameters from section III-B is employed. InFigure 12, average net profit per charging station is depicted,which shows that the proposed framework improves boththe system (in terms of QoS) and its financial performancesignificantly.

VI. TOWARDS A MORE REALISTIC MODEL: THE ROLE OF

COMMUNICATIONS AND INCENTIVES

In this study, we have presented the architecture and anassociated stochastic model for a network of charging stationsfor allocating power and reroute customers in an optimal

5Note that customer profile~ρ is approximately the same since stations arephysically close to each other

Arrv. Rate=20 Arrv. Rate=25 Arrv. Rate=300

20

40

60

80

Net

Pro

fit

Case I

Case IIA

Case IIB

Case III

Fig. 12: Net Profit Comparison

fashion. Deployment of such a network requires the necessarycommunications infrastructure and protocols to ensure timelyinformation dissemination.

Hence, it is important to quantify the impact of communi-cations on the network’s operations. For example, how com-munication delays and losses between EVs, charging stationsand a network coordinator affect routing decisions and henceQoS?

Wireless network technology such as 4G (LTE or WiMAX)and 3G/UMTS, could play a critical role in EV/PHEV roamingschemes. Moreover, in potential “handoff” circumstances,thecommunication network should support inter-grid communica-tions, so that drivers can retrieve up-to-date informationaboutnearest charging stations, available pricing, etc., through theaforementioned smart apps or on-board energy managementsystems [33]. In the latter case, charging stations will bein constant communication with the network coordinator,possibly using a mixture of standard wide area communicationprotocols. In a recent real world application, charging stationsemployed a 2.4 GHz, 802.15.4 full mesh radio protocol [37].Moreover, depending on the volume and the criticality of op-erations, IEC 60870-6 (inter-control center communications)could be employed as well.

The major problem in information dissemination is relatedto the network connectivity, that is, some vehicles could bemissing part of the information since they are temporarilynot connected to the network. Assuming that informationdissemination is provided by means of a specific network,this aspect can be translated into the problem of evaluatingthe degree of connectivity of the employed communicationnetwork.

Another limitation of the proposed framework is that itassumes that EVs strictly adhere to routing decisions takenbythe network coordinator. However, many drivers may deviatefrom the proposed assignments by the network coordinatorand head towards the nearest station. Pricing incentives couldaddress this issues, as discussed in [38].

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